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ABSTRACT: We introduce a new tuberculosis (TB) mathematical model, with state-space variables where are evolution disease states (EDSs), which generalises previous models and takes into account the flux of populations between a high incidence TB country (A) and a community (G) with high percentage of people from (A), plus the rest of the population (C) of a host country (B) with low TB incidence. Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of communities are simultaneously beneficial for the TB control from a global and regional viewpoint. There is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number . Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters that account for the existence of (G), composed of individuals coming from a high incidence area, and the (seasonal) flux between (A) and (B). We also show that the TB transmission rate does not act linearly on , as is common in compartment models where system feedback or group interactions do not occur. Further, we find the most important parameters for the increase of each EDS. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness.

ABSTRACT: We determine the best (smallest) constant in the Anisotropic Sobolev inequality of the form

and the best (smallest) constant in the inequality

where with and . These best constants are obtained by introducing a new method and using variational techniques. The method introduced here seems to have independent interest. We also use this method to find the best constant of the Gagliardo-Nirenberg interpolation inequality involving the -Laplacian

On the Schrödinger-Poisson system with a general indefinite nonlinearity (with L. Huang, J. Chen), Nonlinear Analysis: Real World Applications 28(2016), 1–19.

ABSTRACT: We study the existence and multiplicity of positive solutions of a class of Schrödinger-Poisson systems of the form

where changes sign in , , and the nonlinearity behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that and near , where is the first eigenvalue of in with weight function , whose corresponding positive eigenfunction is denoted by . An interesting phenomenon here is that we do not need the condition

which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. D.G. Costa, H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in , Calc. Var. Partial Differential Equations 13(2001) 159–189.)

ABSTRACT: This paper is concerned with the existence of multiple solutions to the semilinear equation

in a bounded domain of the Heisenberg group with Dirichlet boundary condition, where and is a distance in . By using variational methods, we prove that this problem possesses at least one positive solution and one sign changing solution for some values of and .

ABSTRACT: In this paper, the main problem of study concerns to find a suitable discretization method to (numerically represent) the solution in a Hilbert space, of the general equation

(1)

satisfying a given boundary condition (e.g. Dirichlet), where is a linear differential (or integral) operator and is a nonlinear function with enough regularity. The method Aveiro Discretization Method in Mathematics (ADMM), introduced by Saitoh et al. (2014; doi:10.1007/978-1-4939-1106-6 3), can deal with problem (1) when is a function that do not depends on . In fact, ADMM is a general method for solving by discretization, in a specific optimal sense and when a priori some data may be missing, a wide class of linear mathematical problems by using some key ideas of reproducing kernels and Tikhonov regularization theory. Here, we aim to extend the ADMM method to a more general situation where the nonlinearity may depend on . Then, we apply the scheme to find the (optimal) discretization solution of the problem

ABSTRACT: We study the existence and multiplicity of positive solutions of a class of Schrödinger-Poisson systems of the form

where changes sign in , , and the nonlinearity behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that and near , where is the first eigenvalue of in with weight function , whose corresponding positive eigenfunction is denoted by . An interesting phenomenon here is that we do not need the condition , which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).

where is a positive constant and the nonlinear growth of reaches the Sobolev critical exponent, since for three spatial dimensions. We prove the existence of (at least) a pair of fixed sign and a pair of sign-changing solutions in under some suitable conditions on the non-negative functions , but not requiring any symmetry property on them.

ABSTRACT: We use variational methods to study the existence of at least one positive solution of the following Schrödinger-Poisson system

under some suitable conditions on the non-negative functions and constant , where (critical Sobolev exponent). Note that the nonlinearity involves a critical exponent, the Sobolev embedding is not compact. This will create additional difficulies in the proof of the Palais-Smale condition. We will transform the problem into a nonlocal elliptic equation in , where we also consider the limiting case .

Existence of solutions of sub-elliptic equations on the Heisenberg group with critical growth and singularity (with J. Chen), Opuscula Math. 33:2(2013), 237-254.

ABSTRACT: We consider a class of sub-elliptic equations on the Heisenberg group with a Hardy type singularity and a critical nonlinear growth

where is the -sub-Laplacian with respect to a fixed bases for the corresponding Lie algebra of the Heisenberg group, , is the distance of to the origin given by the homogeneous distance

is the homogeneous dimension, denotes the closure of under the norm and the critical exponent of embedding

We prove the existence of energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

ABSTRACT: We consider the existence of nontrivial solutions of a Dirichlet problem with equation

where is a bounded domain with smooth boundary, is a real continuous and positive function, , (necessarily ), and , with . Note that is the critical Sobolev-Hardy exponent minus 2. By variational methods we show that there are (at least) two positive solutions and (at least) one pair of sign-changing solutions in for any , where is a positive value suitably defined.

ABSTRACT: We consider the existence of nontrivial solutions of the equation

(1)

where is a smooth bounded domain in (). By variational methods and Nehari set techniques, we show that this problem, under some additional hypotheses on , , , and , has four nontrivial solutions in , and that least one of them is sign-changing. Classes of elliptic equations which include (1) have a lost of compactness phenomena, since the nonlinearity has a critical growth imposed by the critical exponent of the Sobolev embedding into . This means that we could not use standard variational methods. On the other hand, due to the presence of the singular term , the problem has a strong singularity at .

ABSTRACT: Mixtures of thiophene with two ionic liquids, namely, and were chosen as prototypes of systems presenting lower critical solution temperature (LCST) and upper critical solution temperature (UCST) behavior, respectively. This distinct behavior is due to different interactions between the constituting species which are investigated here by means of experimental and computational studies. Experimentally, density measurements were conducted to assess the excess molar volumes and and NMR spectroscopies were used to obtain the corresponding nuclear chemical shifts with respect to those measured for the pure ionic liquids. Computationally, molecular dynamics simulations were performed to analyze the radial distribution neighborhoods of each species. Negative values of excess molar volumes and strong positive chemical shift deviations for systems, along with results obtained from MD simulations, allowed the identification of specific interactions between anion and the molecular solvent (thiophene), which are not observed for . It is suggested that these specific -thiophene interactions are responsible for the LCST behavior observed for mixtures of thiophene with ionic liquids.

ABSTRACT: We consider the following quasilinear Schrödinger equation with harmonic potential

(1)

where , is a complex-valued function and is the standard Laplacian operator. We are concerned with stability and instability of standing wave solutions for (1). We will prove the existence of stable standing waves for and the existence of unstable standing waves for (here and after, denotes the critical exponent, i.e. for and for ). Our result indicates that the quasilinear term makes the standing wave more stable than their counterpart in the semilinear case, which is consistent with the physical phenomena and is in striking contrast with the classical semilinear Schrödinger equation.

ABSTRACT: We study the existence of multiple positive weak solutions of the equation

where and () is a bounded domain with smooth boundary, is the critical Sobolev exponent, and . We use variational methods to prove that for suitable , the problem has at least two positive weak solutions.

ABSTRACT: Let be a bounded domain with a -boundary . In this paper we study second order elliptic equations of the form

driven by the Laplacian and -Laplacian differential operators and a nonlinearity which is (-)superlinear (it satisfies the Ambrosetti-Rabinowitz condition). For the -Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. For the semi linear problems using in addition Morse theory, we obtain six nontrivial solutions. We prove seven such multiplicity results. The first five concern problems driven by the -Laplacian, while the last two deal with the particular case (semilinear problems). In all these theorems we also provide precise sign information about the solutions.

A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities (with J. Chen) in Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications 229, 123–137, (2013).

ABSTRACT: We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group of the form

where is the closure of under the norm . Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group, ,

We also point out some extensions to the quasilinear sub-elliptic case.

ABSTRACT: We consider the existence of nontrivial solutions of the equation

where is a smooth bounded domain in (). By variational methods and Nehari set techniques, we show that this equation has at least two nontrivial solutions in , under some additional hypotheses on , , , and , which may be sign-changing. If then the solutions are positive.